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English: The number on the rim of the node circle indicates an order of an element the node represents. Node's colour marks conjugacy class of element with exception that elements of group's center have the same colour (light gray) despite every one of them being of its own conjugacy class of one element.
Order p 2: There are just two groups, both abelian. Order p 3: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p 2 by a cyclic group of order p. The other is the quaternion group for p = 2 and a group of exponent p for p > 2.
In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group. The former may be written as a direct sum of finitely many groups of the form Z / p k Z {\displaystyle \mathbb {Z} /p^{k}\mathbb {Z} } for p {\displaystyle p} prime, and the latter ...
This common order must be a prime number, and the elementary abelian groups in which the common order is p are a particular kind of p-group. [1] [2] A group for which p = 2 (that is, an elementary abelian 2-group) is sometimes called a Boolean group. [3] Every elementary abelian p-group is a vector space over the prime field with p elements ...
Abelians (Latin: Abelonii; also Abelites, [1] Abeloites or Abelonians) were a Christian sect that emerged in the 4th century in the countryside near Hippo Regius in north Africa during the reign of Arcadius. [2] [1] They lived in continence as they affirmed Abel did. They were required to be married but were forbidden to consummate the marriage.
The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table's values are symmetric along its diagonal axis. The group {1, −1} above and the cyclic group of order 3 under ordinary multiplication are both examples of abelian groups, and ...
An object in Ab is injective if and only if it is a divisible group; it is projective if and only if it is a free abelian group. The category has a projective generator (Z) and an injective cogenerator (Q/Z). Given two abelian groups A and B, their tensor product A⊗B is defined; it is again an abelian group.
An abelian group A is torsion-free if and only if it is flat as a Z-module, which means that whenever C is a subgroup of some abelian group B, then the natural map from the tensor product C ⊗ A to B ⊗ A is injective. Tensoring an abelian group A with Q (or any divisible group) kills torsion. That is, if T is a torsion group then T ⊗ Q = 0.